Aptitude skill lesson
Abstract reasoning: skill lesson
Abstract reasoning is rule-finding stripped of language and content: you are given shapes, shading, counts and positions, and asked which rule generates them. Because nothing in the item depends on vocabulary or schooling, it is one of the most widely used constructs in graduate screening, general aptitude batteries and matrix-style tests, and it is the section candidates most often describe as unfair. Usually because they were searching for one rule where the figure encodes three independent ones. The skill is systematic attribute scanning, not flashes of insight. Practice is device-local: no account, and nothing leaves this device unless you export it.
Published · last reviewed
Applies to Novus Learn 0.1.0
What changed, and when
- , Replaced the body of all 27 non-judgement lessons with construct-specific material: six worked examples each, carrying the actual arithmetic, the actual inference or the actual procedure, plus expanded objectives, practice tips and glossary. Every 'Related Learn topics' link now points at a real page on this site rather than a generic search. The eight workplace-judgement lessons are unchanged.
- , Rebuilt the lesson page around a sticky contents rail, per-section links, previous/next lesson navigation, and two graded checkpoints drawn from the open practice bank.
- , Repaired the aptitude integrations behind the lessons so each one links to skill-specific practice instead of the unscoped fixture engine.
- , Published one lesson for each of the 35 aptitude skill constructs: objectives, worked examples, practice tips, glossary, Learn topic links, and sources.
These 35 skill lessons are authored and revised as one set, so they share one revision history rather than 35 identical dates.
Objectives
Copy link- Scan a figure against a fixed attribute checklist (count, shape, size, shading, orientation, position, line style, symmetry) instead of searching for a rule at random.
- Read a three-by-three matrix along rows and columns separately and combine two independent progressions to produce the missing cell.
- Distinguish the three combination rules that look alike on paper (union, intersection and exclusive-or superposition) by testing each against a complete row.
- Track several attributes cycling at different periods within one sequence, and predict a frame that no single attribute determines.
- Tell a rotation from a reflection by checking whether the clockwise order of three marked features is preserved or reversed.
- Use the answer options as evidence, eliminating on one attribute at a time rather than judging each option as a whole.
Checkpoint: does the idea land?
Copy linkBefore the worked examples, check that the objectives above actually landed.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Examples
Copy linkA matrix where the rule is arithmetic on side counts
A three-by-three grid of polygons. The top row runs triangle, square, pentagon. The middle row runs square, pentagon, hexagon. The bottom row runs pentagon, hexagon, and an empty cell. Count sides rather than naming shapes and the structure appears immediately: 3, 4, 5 across the top; 4, 5, 6 across the middle; 5, 6, and the answer across the bottom. Sides increase by one along each row AND by one down each column, so the missing figure has seven sides: a heptagon. Writing the numbers into the cells is the whole technique, and it generalises: whenever a matrix contains countable things, replace every cell with its count before you look for anything else, because a numeric grid makes a progression visible that shapes hide. The distractor set on an item like this is instructive. It will contain a hexagon (correct rule, one step short), an octagon (right idea, overshot), a heptagon with the wrong shading, and a heptagon rotated. Three of those four are only wrong on a second attribute, which is why the answer must be checked on every attribute before you commit rather than on the one that solved the puzzle.
Union, intersection and exclusive-or look identical until you test them
Many matrices build the third cell of each row by combining the first two. Take a row where cell one contains a dot in the top-left corner and a cross in the centre, and cell two contains a cross in the centre and a dot in the bottom-right corner. Cell three contains a dot in the top-left and a dot in the bottom-right, with no cross. That is exclusive-or superposition: elements present in exactly one of the two inputs survive, and elements present in both cancel. If cell three had contained both dots AND the cross, the rule would be union. Everything from both inputs is kept. If it had contained only the cross, the rule would be intersection: only what appears in both survives. All three rules produce plausible-looking figures, so guessing from one row is unreliable. The reliable procedure is to identify a row where the two inputs share at least one element and differ in at least one other, because that is the only configuration where union, intersection and exclusive-or give three different answers. Confirm the rule there, then apply it to the row with the missing cell. Items in this family also hide a fourth variant, where shared elements survive but change colour; catching that one requires checking shading as a separate attribute rather than treating a black cross and a white cross as the same object.
Three attributes on three different clocks
A six-frame sequence. Frame 1: an arrow pointing up, unshaded, with one internal bar. Frame 2: arrow pointing right, shaded, two bars. Frame 3: arrow pointing down, unshaded, three bars. Frame 4: arrow pointing left, shaded, one bar. Frame 5: arrow pointing up, unshaded, two bars. What is frame 6? Take the attributes one at a time. Direction rotates ninety degrees clockwise every frame, a cycle of four, so after up comes right. Shading alternates, a cycle of two, so after unshaded comes shaded. The bar count runs 1, 2, 3, 1, 2, a cycle of three, so after two comes three. Frame 6 is a right-pointing shaded arrow with three bars. The reason this item defeats people is that the whole figure does not repeat until frame 13: the individual attributes come back round every four, two and three frames, but they only come back into phase together after twelve, so the sequence as printed looks like it has no period at all. Treat each attribute as an independent counter with its own cycle length and the difficulty evaporates. The practical habit is to rule three columns on your paper (direction, shading, count) fill them in for every given frame, and extend each column separately before you look at a single answer option.
Odd one out, where the obvious difference is the decoy
Five figures. (a) A square with one diagonal drawn. (b) A triangle with a line from the apex to the base. (c) A pentagon with a line joining two non-adjacent vertices. (d) A hexagon with two lines crossing inside it. (e) A circle with a chord. Which is the odd one out? The difference that jumps out is that (e) is curved and the others are straight-edged, and that is the decoy. It is a real difference, but it is not the one the item is built on. Count enclosed regions instead. Figures (a), (b), (c) and (e) are each divided into exactly two regions by a single line. Figure (d) has two crossing lines and is divided into more than two. The rule is region count, and (d) is the answer. Odd-one-out items reliably plant a salient irrelevant difference (one figure is the only curved one, or the only shaded one, or the only one with a right angle), and hide the operative rule in something countable. The defence is procedural: before choosing, write the value of at least three attributes for all five figures, for example number of enclosed regions, number of straight edges, number of line intersections. The correct answer is the figure that stands alone on exactly one row of that table while the others agree, and if two rows both isolate a figure, the item is ambiguous and you should prefer the countable rule over the categorical one.
Rotation or reflection, settled by clockwise order
A flag-like figure carries three distinguishable marks: a red band, a blue band and a yellow band, which read red, blue, yellow going clockwise around its centre. You are shown a candidate answer figure that also has three bands. Is it a rotation of the original or a reflection of it? Rotating a flat figure in the plane never changes the clockwise order of its features, so any genuine rotation still reads red, blue, yellow clockwise, from whatever starting point you choose. A mirror reflection reverses that order, so a reflected figure reads red, yellow, blue clockwise. Read the cyclic order and the question is answered without imagining any motion at all. This matters because reflections are the standard trap in rotation items: the reflected option looks exactly as plausible as the rotated one, and no amount of mental turning will produce it. The same test works on letters, no rotation of R in the plane produces a backwards R, and on three-dimensional figures, where the equivalent check is whether a right-handed set of three edges at a corner stays right-handed. Note the one honest caveat: a figure with a mirror line of its own is unchanged by reflection, so this test only decides the question for asymmetric figures, which is exactly why item writers use asymmetric ones.
Two dots on the perimeter, moving at different rates
A three-by-three grid where only the eight perimeter cells are used. Number them clockwise starting at the top-left corner: 0 top-left, 1 top-middle, 2 top-right, 3 right-middle, 4 bottom-right, 5 bottom-middle, 6 bottom-left, 7 left-middle. Frame 1 has both dots at cell 0. Frame 2 has one dot at cell 2 and the other at cell 7. Frame 3 has them at cells 4 and 6. Frame 4 has them at cells 6 and 5. What is frame 5? Track them separately. The first dot goes 0, 2, 4, 6, two cells clockwise each frame, so it lands on 8, which wraps to cell 0, the top-left corner. The second goes 0, 7, 6, 5, one cell anticlockwise each frame, so it lands on cell 4, the bottom-right corner. Frame 5 has one dot top-left and one bottom-right. Two counting errors dominate this family. The first is treating the perimeter as nine positions because the grid has nine cells; the centre is not on the path and the perimeter is a loop of eight. The second is assuming both markers share a direction or a speed, which is precisely the assumption the item is testing. When markers move at different rates, the frames where they coincide or sit adjacent are coincidences of the arithmetic, not part of the rule, and reading meaning into them is how a solvable item becomes impossible.
Checkpoint: can you apply it?
Copy linkNow apply it. These come from a later section of the bank, so they are not more of the same.
Two questions from the open practice bank, answered here and scored on this device. Untimed, ungraded, and not added to your practice history, the full bank is where attempts are recorded.
Practice tips
Copy link- Keep a written attribute checklist beside you until it is automatic: count, shape, size, shading, orientation, position, line style, symmetry. Scanning a fixed list is faster than free-associating, and it is what separates candidates who finish the section from those who stall on item four.
- Replace every countable cell with a number before looking for a rule. A matrix of shapes is hard to read; the same matrix written as 3, 4, 5 over 4, 5, 6 solves itself.
- Read matrices along rows and down columns as two separate passes. A large share of items encode one progression in each direction, and a rule that only works across rows is usually half the answer.
- Use the options as data. Sort them by one attribute, discard the group that cannot be right, then sort the survivors by the next. Eliminating on single attributes is far more reliable under time pressure than judging whole figures.
- Check the cyclic order of three features before accepting any rotation option, and remember that a reflected asymmetric figure can never be produced by turning the original in the plane.
- Practise untimed until you can state the rule in one sentence for every item you answer, including the ones you got right by feel. Attempts are stored on this device only, so an unhurried pass where you narrate each rule aloud costs nothing and is what actually builds the scanning habit.
Glossary
Copy link- Attribute
- A single independently varying property of a figure, such as shading, count or orientation. Most difficult items vary three or more attributes at once, each on its own cycle.
- Matrix item
- A grid of figures, usually three by three, with one cell missing. Rules typically run along rows, down columns, or both at once.
- Superposition
- A rule that builds one figure by combining two others. The three common variants keep the union of elements, the intersection, or the elements appearing in exactly one input.
- Exclusive-or rule
- The superposition variant in which elements common to both inputs cancel and only the unshared elements survive. Distinguished from union and intersection by testing a row whose inputs both share and differ in elements.
- Distractor
- An incorrect answer option built to match the correct one on the attribute that solved the item while differing on another. It is why a figure must be checked on every attribute before selection.
- Chirality
- The handedness of an asymmetric figure. Rotation preserves it and reflection reverses it, so the clockwise order of three marked features tells the two transformations apart.
- Cycle length
- The number of frames after which an attribute repeats. When several attributes have different cycle lengths, the whole figure only repeats after their least common multiple.
- Enclosed region
- An area fully bounded by lines within a figure. It is one of the most common hidden rules in odd-one-out items because it is countable and visually unobtrusive.
Sources
Copy link- Every figure, matrix and sequence described above was constructed and verified for Novus Learn. Side counts, superposition outcomes, attribute cycle lengths and perimeter positions were each checked by hand.
- Matrix-style non-verbal items are a long-established public assessment format; this lesson describes the format in general terms and reproduces no item from any published test.
- Terminology checked against the public Wikipedia articles 'Chirality', 'Exclusive or' and 'Least common multiple'. Definitions only.
- Novus Learn aptitude construct registry (catalog seed) for the construct scope and related suite mapping.
- Public educational framing only: not affiliated with any official exam board, publisher or employer, and no copyrighted test item is reproduced.
Your notebook for this lesson
Saved sections and notes are kept in this browser, on this device. No account is required, and nothing is sent to us unless you create one and switch on backup. They travel with your backup file from My Learning, and they are lost if you clear this site’s data.
Saved sections
None yet. Use Save beside any section heading above to keep it here.
4000 characters left
Your own words, so you can find this again from any other lesson. Up to 12.